Gis Spatial Data Structures

Jörg-Rüdiger Sack

School of Computer Science, Carleton University

Ottawa, Canada K1S 5B6, sack@scs.carleton.ca

95.5204 :

Spatial Data Structuresfor GIS

Geometric Objects

A geometric object is an object which characterizes a geometric component, i.e., the

• location and

• shape

of the object in space.

In addition, there is the attribute component which we will ignore for the discussion in this chapter.

Example

Planar subdivisions for example are collections of polygons which represent towns or municipality regions.

The geometric information about the location of the place is stored through the polygon.

(Non-geometric information such as name, size, …. are also stored.)

Operations

There are many operations that need to be carried out on geometric objects, these include:

• point in polygon (point location)

• traversal of a subregion (window queries)

• intersection tests

• ….

• other operations include:

– distance, containment, intersection

Operations cont’d

1. objects are stored on disc examining, i.e., retrieving all objects is extremely inefficient!

2. checking each object is time-consuming (even after retrieval) as the geometry may be complex.

Idea: support spatial queries to geometric objects by realizing a filter, i.e., providing a superset of the solution set and subsequently refine that set to the correct solution.

Filter

Sometimes this approach is referred to as

coarse filter

fine filter

where coarse filter refees to the retrieval of a subset of adjacent objects

followed by the fine filter which analyzes geometric properties of the objects.

The Idea of a Filter

Create a bounding box for 2-d geometric objects.

Bounding box: = smallest axis parallel rectangle containing the geometric object

The database search key for the geometric object is now that of the bounding box.

There are many data structures for multi-dimensional

For d dimensional objects, let Ui = universe in the ith dimension. Then

U = U1x U2 x U3 … x Ud is the d-dimensional universe containing all geometric objects.

Filter cont’d

G : be a particular set of geometric objectsg ε G described as:

– g.b d-dim bounding box– g.rest other attributes that are not relevant for the search

g = (b, rest)

b= (l1, r1, l2, r2,…, ld, rd) d-dim interval

[l1, r1] x … x [ld, rd] where b.li : left and r.ri is the right interval boundary of the ith interval.

we use: g. li for g.b. li and g. ri for g.b. ri

Example

The Task

Task: find a secondary storage structure S supporting the following operations:

(1)Range query

(2) Search

(3) Insert

(4)Remove (delete)

more formally next

Rangequery

Rangequery (w, S(G))

range w, G is stored in S

report all objects g in G with g.b ∩ w ≠ Ø

assumption: two rectangles that only intersect at a boundary do not intersect, i.e.,

intersection (A,B) := closure (interior of A ∩ interior of B)

Rangequery cont’d

reports: 1, 6, 3, 5

Search

Search (b, S(G))

for bounding box b and G stored in S

report all objects g in G with g.b =b

search - example

the object g (blue)has bounding boxmatching the querybox

Search

Insert (g, S(G))

S(G) := S(G U {g}) add g to G and store it in S

Remove (Delete)

Remove (Delete) (b, S(G))

remove object g is g.b = b andS(G) := S(G \ {g}) remove g from G and store the result

Comments

1. While uniqueness is somewhat the underlying assumption it does not pose any serious implementation difficulties.

2. For insert, search and delete

the key is spatial, but

the spatial location is not referenced

-> this can be handled by traditional secondary data structures such as B-trees, dynamic hashing, …

e.g., map the 2d key components into one 1-dimensional key (lexicographic)

Comments

Thus searchers can be handled!

Problem: Queries of type Rangequery

they are space relevant and the above storage schemes show serious deficiencies

Objective

Find data structure for geometric objects such as points, polygons etc that allow efficient retrieval.

Primary concern:

When accessing data, long chains of pointers that are crossing disk block boundaries must! be avoided.

Game: design data structures with

– small internal memory access structure

– efficient dynamically updates

Basic Concepts

Basic Concepts for spatial structures

access time: DRAM (dynamic random access memory) chips for personal computers have access times of 50 to 150 nanoseconds (billionths of a second).

Fast hard disk drives for personal computers boast access times of about 9 to 15 milliseconds.

Note that this is about 200 times slower than average DRAM.

Basic Concepts

Actually many machines have even larger ratios than that.

Typical numbers are:

Memory access time (seconds): 10-7 … 10-6

Disc access time (seconds): 10-2 … 10-1

ratio disc/memory access time: 104 … 105

Basic Concepts

Typical size of transfer unit (bits):

Memory : 10 … 102

Disc : 104 … 105

ratio disc/memory transfer size: 102 … 103

Basic Concepts

The time for an operation is thus determined by the time to retrieve the data + the time required to carry out the local computation.

For many operations, # of disc accesses is the dominating factor. However, there are geometric problems where also the internal computations are costly.

Objective

Find data structure for geometric objects such as points, polygons etc that allow efficient retrieval.

Primary concern:

When accessing data, long chains of pointers that are crossing disk block boundaries must! be avoided.

Game: design data structures with

– small internal memory access structure

– efficient dynamically updates

Proximity

Data on discs are seen to be organized in BLOCKS.

A block is a unit of data that is retrieved in one shot from a disc.

A block contains many data, these should be useful for the algorithm and its execution,.

1. local maintenance of proximity; i.e, physically close in space

2. global maintenance of proximity; objects stored in adjacent blocks are physically close.

Proximity

especially the last points is very difficult to obtain.

There is no perfect data organization!

Even small improvements in that, yield accelerations that are noticeable.

Central issue

Organizing the embedding space versus organizing its content.

We will discuss data organizations who are dependent on the data and mostly those who are dependent on the space.

This is the key distinction between space and non-spatial data structures.

Non-spatial data structures

Data structures for non-spatial data any search structure that you may have encountered for example: binary search tree.

•searches are comparative:

•structures exist and are readily available also balanced

– AVL, 2-3 trees, red-black trees

excellent search structures also for statistical queries including median, percentiles,

Non-spatial data structures

Such data structures are not designed for, nor can they efficiently handle:• general location queries

– nearest neighbour

– identify clusters in data

Review of address computation schemes

1. Hashing

2. radix trees

3. tries

these assign an address of a storage cell to any key value x

(course notes)

k-d trees

k-d trees were invented by Bentley ’75

as generalizations of search trees i.e. comparative

other relevant structures:

Lueker 78, Lee&Wong ’77, Willard’78, Bentley’79, Bentley and Maurer’80

k-d trees

An example:

x : 50

y : 15 y : 4

k-d trees

Problems:

• it is hard to balance these structures, i.e., get log height

• 1-d is easy

• space partitioning created lacks regularity

• difficult neighbour queries

First approaches

First approaches to spatial data structures

• based on the existing search structures

• data stored!

• not the space in which the data was embedded

filter illustration fora rectangular space partitioning

query q

drop ignored

querycells

report allobjects thatintersect q

the oval is examinedand then droped

not retrieved

Comment

Spatial data structures cover the space with cells.

Each cell is stored on disc and therefore is associated with a disc block or blocks.

Three-phase model

Three steps:1. Cell addressing

for a given query find all “cells” of the partitiongthat could contain elements relevant to query

2. Coarse filterretrieve the elements found in Step 1 from disc

3. Fine Filterexamine the elements (Step 2) if they fit the query

Tree-based schemes

Work has been done on the internal memory data structures: segment trees and range trees

and how they can be extended external storage.

This is not covered here. Could be a good topic for a class presentation.

Three philosophies

1. Space driven: 1. multi-dimensional linear hashing,2. space filling curves3. ...

2. Data driven1. k-d-B-trees2. ….

3. Combinations1. grid file and its variants2. Bang file, ….

Linear hashing

viewed as a spatial data structurepartition the 1-d data space into intervals

interval sizes half of previous; simple addressing scheme

36 754

doubling

Doubling is typically adding a bit to the front (or back) of the string created thus far.

e.g., in some of the schemes you would see

0 1 00 10 01 11

addedbit

this means that when you run out of space a piece of the same size is appendedresulting in a doubling of the space used. However address calculations are simple!

MOLPHE

Multidimensional Order Preserving Linear Hashing

2 23 3

Note the alternation of split in the dimensions. 1st split by x; 2nd split by y; 3rd split again by x-axis. Note also the each block is split.

z-hashing

Dynamic z-hashing

Note the addressing function is different to the one given above. The reason is that proximity is better maintained between adjacent blocks.

space-filling curves

The above schemes define a traversal of the space.

Here we list other space filling curves that are typically used.

They have different properties and studies have been carried out on them.

E.g., Peano, z-ordering and Hilbert

space-filling curves

Hilbert

Z-orderG.M. Morton

Z-order

z-order of a point withcoordinate x,yis obtain by bit-wise interleaving ofthe x and y bits.

Ex.:y = 2 = 010x = 5 = 101

25 = 0 1 1 0 0 1

Z-order

z-order of a point withcoordinate x,yis obtain by bit-wise interleaving ofthe x and y bits.range queriesare possibleslight care needs to be taken to find successors of point in z-order

Hilbert curve: maping

range queriesmore natural,but successorfunction moredifficult thanwith z-ordering.

Hilbert curve cont’d

direction in whichto draw the elementsof the Hilbnert curve

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